The given series is:

$8 + 6 + \frac{9}{2} + \frac{27}{8} + \dots$

We need to determine if the series converges or diverges, and if it converges, find its sum.

Step 1: Identify the type of series

This appears to be a geometric series. A geometric series has the form:

$a + ar + ar^2 + ar^3 + \dots$

where $a$ is the first term, and $r$ is the common ratio between consecutive terms. Let's check if there's a common ratio between the terms.

• First term $a = 8$.
• Second term $6 = 8 \times r$, so $r = \frac{6}{8} = \frac{3}{4}$.
• Third term $\frac{9}{2} = 6 \times r$, so $r = \frac{\frac{9}{2}}{6} = \frac{9}{12} = \frac{3}{4}$.
• Fourth term $\frac{27}{8} = \frac{9}{2} \times r$, so $r = \frac{\frac{27}{8}}{\frac{9}{2}} = \frac{27}{8} \times \frac{2}{9} = \frac{3}{4}$.

Thus, the common ratio $r = \frac{3}{4}$, confirming that this is a geometric series.

Step 2: Test for convergence

A geometric series converges if the absolute value of the common ratio $|r| < 1$. In this case:

$\left| \frac{3}{4} \right| = \frac{3}{4} < 1$

So, the series converges.

Step 3: Find the sum of the series

The sum $S$ of an infinite geometric series is given by the formula:

$S = \frac{a}{1 - r}$

where $a$ is the first term and $r$ is the common ratio. Substituting the values:

$S = \frac{8}{1 - \frac{3}{4}} = \frac{8}{\frac{1}{4}} = 8 \times 4 = 32$

Thus, the sum of the series is 32.

Would you like more details or have any questions? Here are some related questions to explore:

1. How do you test for convergence in other types of series?
2. What happens to a geometric series when the ratio $r$ is greater than or equal to 1?
3. Can a geometric series have negative terms and still converge?
4. How is the sum formula for a finite geometric series derived?
5. What are some real-world applications of geometric series?

Tip: In a geometric series, when $|r| < 1$, each successive term becomes smaller and smaller, ensuring the series converges.